Factorial question

Question

How you solve something like 2/20! without a calculator?

Answer 1

For this particular problem, you would write:

 

2/20! =

 

2 / (20×19×18×17×16×15×14×13×12×11×10×9×8×7×6×5×4×3×2×1)

 

You can do a bit of simplifying, since there's a 2 on the top and on the bottom that cancel out. Also, anything times 1 is itself. So, you wind up with:

 

1 / (20×19×18×17×16×15×14×13×12×11×10×9×8×7×6×5×4×3)

 

Really, all you can do after that is multiply the denominator out by hand. I would be shocked if you were asked to do that, though, since that's a huuuuuge number! Something like 1.2×10¹⁸.

 

 

You'd be more likely to be asked to do something like one these by hand:

 

3072/10!  or  10!/15!

 

Would seeing one or both of those worked out help more?

 

 

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3072/10! works a bit like 10!/15!. Our main goal is going to be to cross out as many like factors as we can from the top and the bottom, then solve whatever's left over.

 

Start by factoring 3072:

 

3072 = 3×1024 = 3×2×512 = 3×2×2×256 = 3×2×2×2×128 = 3×2×2×2×2×64 = ... = 3×2¹⁰

 

So, in the numerator, we have a 3 and ten 2s.

 

The denominator, written out, would be:

 

10! = 10×9×8×7×6×5×4×3×2×1

 

Immediately, you can get rid of that 1 in the denominator. You can also cross out the 3 in the numerator and in the denominator and one pair of 2s. That leaves:

 

2⁹ / (10×9×8×7×6×5×4)

 

Next, notice that 2⁹ = (2²)(2³)(2⁴) = 4×8×2⁴:

 

(4×8×2⁴) / (10×9×8×7×6×5×4)

 

Now you can cross out the 4s and 8s, leaving you with:

 

2⁴ / (10×9×7×6×5)

 

There are still a couple of 2s hiding in the denominator. 10 = 2×5 and 6 = 2×3, so you have:

 

2⁴ / (2×5×9×7×2×3×5)

 

Cross out two more 2s from the top and the bottom, leaving you with:

 

2² / (5×9×7×3×5)

 

Now there's nothing left to cross out, but the rest is manageable. Multiply 9×7 = 63 and 5×5×3 = 25×3 = 75 to get:

 

4 / (63×75)

 

You can finish calculating the denominator by hand to get:

 

4 / 4725

 

Hope this helps!

Answer 2

The short answer would be, by hand. You can try multiplying the numbers that are easier together, but eventually you will have to work everything out.

 

Unless, of course, you can submit the answer as is. On a test without a calculator, 2/20! is a perfectly acceptable answer.